Assignment 1: Blackbody Radiation

Assignment 1: Blackbody radiationDue date: 7 September 2009, 11 amAssignment 1: Blackbody Radiation1. If the absolute temperature of a black body is increased from T to 1.414T , by whatfactor is the total emitted power per unit area (R T ) increased? (Use Stefan’s law.)2. Derive the value of the constant A in the Wien’s displacement law,λ max T = A, (1)where λ max as the wavelength at which the spectral radiance is maximum.3. A particular radiating cavity has the maximum of its spectral distribution of radiatedpower at a wavelength of 27 µm (Infrared Region). The temperature is then changedso that the total power radiated by the cavity is 16 times higher. At what wavelengthdoes the new spectral distribution have its maximum value? (Use Stefan’s and Wien’sdisplacement laws.)4. Find the energy density of black body radiation at T = 6000 K in the range from 450to 460 nm, assuming that this range is so narrow that the energy density functionρ T (f) does not vary much over it. (Use Planck’s radiation law.)5. (a) Show that the expression for the number of standing waves can also be expressedin terms of wavelength and written as,N(λ) dλ =8πV dλλ 4 , (2)where V = L 3 is the volume of the cubic cavity.(b) How many independent standing waves with wavelengths between 9.5 and 10.5 mmare present in a cubical cavity 1 m on a side?6. Assuming that the energies of the radiation modes inside a cavity are discrete andequal to E n = nhf, prove the relationship,∑ ∞n=0 E n p(E n )∑ ∞n=0 p(E =n)where p(E) is the Boltzmann distribution function.hfexp (hf/kT ) − 1 , (3)Chapter 1 of Eisberg and Resnick. 1

Assignment 1: Blackbody radiationDue date: 7 September 2009, 11 am7. Show that for very large wavelengths λ, Plank’s formula for spectral radiance,reduces to the Rayleigh-Jeans formula,ρ (λ) = 8πhcλ 5 1exp (hc/λkT ) − 1 , (4)ρ(λ) = 8π kT. (5)λ4 (Note that Rayleigh-Jeans or Plank’s formulas can express the spectral densities eitherin terms of frequency or wavelength. Refer to the approach followed in Question 5.)8. A tuning fork has a natural frequency of 500 Hz and its total vibrational energy is0.04 J. An atom emits orange light at a frequency of 5.00×10 14 Hz and has an energy of2.0 eV. Which of these systems is more likely to exhibit quantum graininess? Supportyou answer quantitatively.Chapter 1 of Eisberg and Resnick. 2